document.write( "Question 159386This question is from textbook pure core maths1 & 2
\n" ); document.write( ": Hi, i'm not sure which topic it should go in but anyway...\r
\n" ); document.write( "\n" ); document.write( "i am struggling to find out how to get the answer to this.\r
\n" ); document.write( "\n" ); document.write( "'Find the gradient of the following curves at the given point'\r
\n" ); document.write( "\n" ); document.write( "y = 3*rootx(x-2)^2\r
\n" ); document.write( "\n" ); document.write( "at the point where x = 3\r
\n" ); document.write( "\n" ); document.write( "i have got the equation to
\n" ); document.write( "y = 3*rootx(x^2 -4x + 4)
\n" ); document.write( "i then have to find the gradient (dy/dx) and substitute in x (3)\r
\n" ); document.write( "\n" ); document.write( "but i can't get the right answer, which the book says is 29*root3\r
\n" ); document.write( "\n" ); document.write( "thanks in advance. \n" ); document.write( "
Algebra.Com's Answer #117533 by Fombitz(32388)\"\" \"About 
You can put this solution on YOUR website!
First off, let's do a substitution.
\n" ); document.write( "Let
\n" ); document.write( "\"u=x%28x-2%29%5E2\"
\n" ); document.write( "then
\n" ); document.write( "\"y+=+3%2Au%5E%281%2F2%29\"
\n" ); document.write( "\"dy%2Fdx=3%2A%281%2F2%29u%5E%28-1%2F2%29%2A%28du%2Fdx%29\"
\n" ); document.write( "Now we can go back and substitute,
\n" ); document.write( "Let's look at u,
\n" ); document.write( "\"u=x%28x-2%29%5E2\"
\n" ); document.write( "\"u=x%28x%5E2-4x%2B4%29\"
\n" ); document.write( "\"u=x%5E3-4x%5E2%2B4x\"
\n" ); document.write( "\"du%2Fdx=3x%5E2-8x%2B4\"
\n" ); document.write( "\"u%283%29=3%283-2%29%5E2=3%281%29=3\"
\n" ); document.write( "\"%28du%2Fdx%29%283%29=3%283%29%5E2-8%283%29%2B4=27-24%2B4=7\"
\n" ); document.write( "Substitute all those values into dy/dx,
\n" ); document.write( "\"%28dy%2Fdx%29=3%2A%281%2F2%29u%5E%28-1%2F2%29%2A%28du%2Fdx%29\"
\n" ); document.write( "\"%28dy%2Fdx%29%283%29=3%2A%281%2F2%29%283%29%5E%28-1%2F2%29%2A%287%29\"
\n" ); document.write( "\"%28dy%2Fdx%29%283%29=%2821%2F2%29%283%29%5E%28-1%2F2%29=%2821%2F2%29%281%2Fsqrt%283%29%29\"
\n" ); document.write( "\"%28dy%2Fdx%29%283%29=%2821%2F2%29%28sqrt%283%29%2F3%29=%287sqrt%283%29%29%2F2\"
\n" ); document.write( "Is this the answer you got??
\n" ); document.write( ".
\n" ); document.write( ".
\n" ); document.write( ".
\n" ); document.write( "We can check the answer by calculating dy/dx near 3 using x and y values.
\n" ); document.write( "Let's use
\n" ); document.write( "x=2.99, then \"y=+3%2Asqrt%282.99%282.99-2%29%5E2%29=+5.13561\"
\n" ); document.write( "x=3.00, then \"y=+3%2Asqrt%283%283-2%29%5E2%29=+5.196152\"
\n" ); document.write( "x=3.01, then \"y=+3%2Asqrt%283.01%283.01-2%29%5E2%29=+5.256854\"
\n" ); document.write( "then we can calculate the derivative on the left of 3 and the right of 3 and take the average.
\n" ); document.write( "(dy/dx)left=y(3.00)-y(2.99))/(3.00-2.99)=(5.196152-5.13561)/(0.01)=6.054233
\n" ); document.write( "(dy/dx)right=y(3.01)-y(3.00))/(3.01-3.00)=(5.256854-5.196152)/(0.01)=6.07011
\n" ); document.write( "We take the average of those two and get
\n" ); document.write( "(dy/dx)(3)=6.062
\n" ); document.write( "Let's look at the value we calculated,
\n" ); document.write( "\"%287%2Asqrt%283%29%29%2F2=12.12436%2F2=6.062178\"
\n" ); document.write( "Looks like the two match.
\n" ); document.write( ".
\n" ); document.write( ".
\n" ); document.write( ".\r
\n" ); document.write( "\n" ); document.write( "Meanwhile, the answer you provided doesn't look close.
\n" ); document.write( "\"29%2Asqrt%283%29=+50.22947\" is not near the answer we calculated and verified.
\n" ); document.write( "Check the answer again. \n" ); document.write( "
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